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1 Math 140 Spring 2017 Suggested Final Review Problems 1. Is each of the following statements true or false? Explain. (a) If f(x) = x 2, then f(x + h) = x 2 + h 2. (b) If g(x) = 3 x, then g(x) can never be zero (c) The range of f(x) = 1 x is all real numbers. (d) The domain of f(x) = x x is all real numbers (e) The independent variable in an exponential function is always found in the exponent. (f) If your salary, S, grows by 4% each year, then S = S 0 (0.04) t, where t is in years. (g) If we are given two data points, we can find a linear function and an exponential function that go through these points. (h) ln(ab t ) = t ln(ab) (i) ln(1/a) = ln a (j) ln a ln b = ln(a + b) (k) ln a ln b = ln a ln b. (l) As x grows through large positive values, y = x + 18 approaches y = 2. x + 9 (m) The rational function f(x) = x + 2 x 2 has a zero at x = 2. 4 (n) In general, the rational function r(x) = p(x) must have at least one zero. q(x) (o) Rational functions never cross an asymptote. (p) A fourth degree polynomial can have two turns. (q) Polynomials have no asymptotes. (r) The function 2 x dominates the function 100x Let f(x) = x + 2 x. Find the domain of this function, and solve f(x) = In month t = 0, a small group of rabbits escapes from a ship onto an island where there are no rabbits. The island rabbit population, p(t), in month t is given by p(t) = (0.9) t, t 0. (a) Evaluate p(0), p(10), p(50) and explain their meaning in terms of rabbits. (b) Graph p(t) for 0 t 100. Describe the graph in words. Does it suggest the growth in population you would expect among rabbits on an island? (c) Estimate the range of p(t). What does this tell you about the rabbit population? (d) Explain how you can find the range of p(t) from the formula. 4. From the following scenario, sketch a graph, and then discuss the domain and the range of the function. Label the axes on your graph. 1

2 Ice is removed from a freezer, and heat is applied. The ice warms to melting temperature at a constant rate. It stays at this melting temperature until it is completely melted. It warms again at a constant rate until it reaches boiling temperature. The temperature remains constant until the water completely boils away. 5. A taxi company charges two dollars for the first mile then 20 cents for each succeeding tenth of a mile. Express the cost C (in dollars) od a ride as a function of the distance travelled in miles and draw the graph of this function. 6. Classify each function (power, exponential etc.): f(x) = 5 x; g(x) = 1 x 2 ; h(x) = x 9 + x 4 ; r(x) = x2 + 1 x 3 + x ; y = 10x ; y = x The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 miles and in June it cost her $460 to drive 800 miles. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the y intercept represent? (e) Why does a linear function give a reasonable model in this case? 8. Explain how the following graphs are obtained from the graph of y = f(x). y = 5f(x); y = f(x 5); y = f(x); y = 5f(x); y = f(5x); y = 5f(x) Graph the following functions (without a calculator, or plotting points): y = 1 ; y = 2 + cos x; x y = cos(x/2); y = x 2 + 2x + 3; y = 1 3 sin(x π 6 ); y = x Find the functions f g, g f, f f, g g and their domains, if f(x) = 2x 2 x g(x) = 3x + 2 f(x) = 1 x g(x) = x3 + 2x f(x) = 1 x Find a formula for the inverse of the following functions. g(x) = x 1 x + 1. f(x) = 1 + 3x 5 2x ; f(x) = 5 4x3 ; f(x) = 2 + 5x. 2

3 12. Solve for x: e x = 5; ln x = 2; e x2 = The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is 100,000 P (t) = e t where t is measure in years. (a) Graph this function and estimate how long it takes for the population to reach 900. (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for the population to reach 900. Compare with the result in part (a). 14. Convert from degrees to radians: 210 ; 36 ; Convert from radians into degrees: 4π; 5π 12 ; Find the remaining trigonometric functions if sin θ = 3 5, 0 < θ < π Solve the following equations: sin 2x = cos x; tan x = 1; 2 cos x 1 = 0; 3 cot 2 x = 1; 2 sin 2 x = 1; 2 + cos 2x = 3 cos x. 18. Find the exact value of the following expressions: 3 sin 1 ( 2 ); cos 1 ( 1); arctan( 1) 19. Solve the following equations, if 0 x 2π: (a) cos 2 x + 2 sin 2 x = 1.75 ; (b) 2 cos 2 x + 3 sin x = 0 ; arcsin 1; sin(sin 1 (0.7). 0 x 2π 2π x 2π 20. Find all the zeroes and asymptotes of the following rational functions. Describe all asymptotic behavior of the functions. a. 2x x 2 1 b. 1 4x2 x c. x3 11x x 36 2x 2 x 1 d. 2x x 3 + 7x 2 + 2x Find the difference quotient f(x + h) f(x) h if f(x) = x 2 + 3x. 3

4 22. In triangle ABC, angle A = and, sides b and c are 3 cm and 7 cm respectively. Find the missing sides and angles. 23. Solve the equation 2 log(x + 2) + log 4 = log x + 4 log Show the following trigonometric identities: (a) csc x sin x = cos x cot x (b) cos x csc x + sin x sec x = 2 csc 2x 25. The temperature, H, in F, of a cup of coffee, t hours after it is set to cool is given by the equation H = (1/4) t. (a) What is the coffee s temperature initially? After 1 hour? After 2 hours? (b) How long does it take for the coffee to cool down to 90 F? 26. Solve for x: log x + log(x 1) = log 2. Would you get a different value for x if instead the equation was ln x + ln(x 1) = ln 2? 27. A population P (t) (in millions) in year t increases exponentially. Suppose P (8) = 20 and P (15) = 28. (a) Find a formula for P (t) without using e. (b) If P (t) = ae kt, find k. 28. Oil leaks from a tank. At hour t = 0 there are 250 gallons of oil in the tank. Each hour after that, 4% of the oil leaks out. (a) What percent of the original 250 gallons has leaked out after 10 hours? Why is it less than 10 4% = 40%? (b) If Q(t) = Q 0 e kt is the quantity of oil remaining after t hours, find the value of k. What does k tell you about the leaking oil? 29. Give the meaning and units of the inverse function: (a) V = f(t) is the speed in km/hr of an accelerating car, t seconds after starting. (b) I = f(r) is the interest earned, in dollars, on a $10,000 deposit at an interest r% per year, compounded annually. 30. Show that the functions f(x) = 1 x 2 and 1 x + 2 are inverse to each other. 31. Find the inverses of the following functions: (a) f(x) = 3 + x 4, (b) f(x) = 7 3x 5 4x. 32. If T (x) = x+1 2x, write the expression for: T (y 2); T (2 y); y T (2); T (y) T (2). 33. The cost (in dollars) of manufacturing x units per day (for 0 x 90) is given by C(x) = 0.25x x The revenue (in dollars) generated by x is given by R(x) = 55x. The profit (or loss) generated by x units is the difference between the revenue generated by x units and the cost of manufacturing x units. 4

5 (a) How many units can be manufactured for a cost of $3075? What is the revenue generated at this level of production? (b) Determine a function P that represents the profit (or loss) generated by x units. (c) Determine the zero of P in part (b). Explain the significance of this value. 34. A small café sells coffee for $0.95 per cup. On average, it costs the cafe $0.25 to make a cup of coffee (for grounds, hot water, filters). The café also has a fixed daily cost of $200 (for rent, wages, utilities). (a) Let R, C and P be the café s daily revenue, costs and profit, respectively for selling x cups of coffee in a day. Find formulas for R, C and P as a function of x. (b) Plot p against x. For what x-values is the graph of P below the x-axis? Above the x-axis? Interpret your results. (c) Interpret the slope and both intercepts of your graph in practical terms. 35. Determine which of the functions in the table below could be linear and which could be exponential. Write formulas for the linear and exponential functions. x f(x) g(x) h(x) Caffeine is a chemical stimulant that is found in coffee and cola. A typical human body eliminates 10% of this compound each hour after ingestion. (a) Write a function Q that models the amount of caffeine present in the body t hours after the ingestion, given that Q 0 is the amount ingested. (b) Suppose that Maria has a double espresso coffee (60 mg of caffeine) at 9am just before precalculus lab. How much coffee remains in her system after the lab is over at 11am? 37. In 1999, the world s population reached 6 billion and was increasing at a rate of 1.3% per year. Assume that this growth rate remains constant. (a) Write a formula for the world population (in billions) as a function of the number of years since (b) Use your formula to estimate the population of the world in year (c) Sketch a graph of world population as a function of years since Use the graph to estimate the doubling time of the population of the world. 38. Let f be a piecewise-defined function given by 2 x x < 0 f(x) = 0 x = x x > 0. (a) Graph f for 3 x 4. (b) The domain of f is all real numbers. What is its range? (c) What are the intercepts of f? (d) What happens to f(x) as x and x? 5

6 39. What is the balance after 1 year if an account containing $500 earns the yearly nominal interest of 5%, compounded (a) annually (b) weekly (c) every minute (525,000 per year) (d) continuously? 40. Decide whether the following functions could be linear, exponential or are neither. For those that could be linear or exponential, find a formula. x f(x) x g(x) x h(x) Accion is a non-profit microlending organization which makes small loans to entrepreneurs who do not qualify for bank loans. A New York woman who sells clothes from a cart has the choice of a $1000 loan from Accion to be repaid by $1160 a year later and $1000 loan from a loan shark with an annual interest rate of 22/compounded annually. (a) What is the annual interest rate charged by Accion? (b) To pay off the loan shark for a year s loan of $1000, how much would the woman have to pay? (c) Which loan is a better deal for the woman? Why? 42. Find the annual growth rate of a quantity which (a) doubles in size every 7 years (b) triples in size every 11 years (c) grows by 3% per month (d) grows by 18% every 5 months. 43. Let f(x) = 2 x. On the same set of axes, draw the following graphs indicating all important features: a. y = f(x) b. y = f(x 5) c. y = f(x) + 5 d. y = f(2x) e. y = 2f(x) f. y = f( x) 44. For t 0, let H(t) = (0.91) t give the temperature of a cup of coffee in degrees Fahrenheit t minutes after it is brought to class. (a) Find formulas for H(t + 15) and H(t) (b) Graph H(t), H(t + 15) and H(t) (c) Describe in practical terms a situation modelled by the function H(t + 15). What about H(t) + 15? (d) Which function, H(t + 15) or H(t) + 15 approaches the same final temperature as the function H(t)? What is that temperature? 6

7 45. Find the equation of the parabola with vertex (1, 3) that passes through the point (3, 11). 46. A ballet dancer jumps in the air. The height, h(t), in feet, of the dancer at time t, in seconds since the start of the jump, is given by h(t) = 16t T t, where T is the total time in seconds that the ballet dancer is in the air. (a) Why does this model apply only for 0 t T? (b) When, in terms of T, does the maximum height of the jump occur? 47. Express the quadratic function f(x) = x 2 8x + 8 in standard form, sketch its graph and find its maximum/minimum value. 48. The angle of elevation to the top of the Empire State Building in New York is found to be 11 from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the Empire State Building. 49. A company in a northern climate has sales of skis as given by: S(t) = 10[1 cos(π/6)t], where t = time, in months since July 1, and S(t) is in thousands of dollars. (a) Sketch a graph of the function over a 12-month interval [0,12]. (b) What is the period of the function? (c) What is the minimum amount of sales and when does it occur? (d) What is the maximum amount of sales and when does it occur? 50. A patient in the hospital had an illness in which his temperature (in degrees Celsius) varied from a low of 37 to a high of The length of time between successive highs is 16 days. 51. Solve: (a) Determine the formula for the temperature, T, of the patient at time t in days since the beginning of the illness. Assume that the function describing the temperature can be modeled with a sine function, with no phase shift. (b) Sketch a graph of the function over the interval [0, 20]. (c) According to this model, what was the patient s original temperature? (d) What is the patient s temperature on day 4 of his illness? (e) When does the first temperature high occur? (a) sin x = 1/2 (b) 2x(sin x) = x (c) 2 cos 2 x = 1 cos x 52. Find f(f(1)) for 2 x 0 f(x) = 3x < x < 2 x 2 3 x Let f(x) = 1/x. For n a positive integer, define f n (x) as the composition of f with itself n times. For example, f 3 (x) = f(f(f(x))). 7

8 (a) Evaluate f 7 (2). (b) Evaluate f 24 (5). Do you see a pattern? 54. Write the function y = sin(x 3 ) as a composition of two functions. 55. Decompose g(x) = (x 2 + 2) 3 into two or three functions. 56. If C = f(q) gives the cost, in dollars, to manufacture q items, what does f 1 (C) represent? What are its units? 57. Let P = f(t) = 10e 0.02t give the population in millions at time t in years. Find and interpret f 1 (P ). 58. There is a linear relationship between the number of units,n(x), of a product that a company sells and the amount of money, x, spent on advertising. If the company spends $25,000 on advertising, it sells 400 units, and for each additional $5,000 spent, it sells 20 units more. (a) Calculate and interpret N(20, 000). (b) Find a formula for N(x) in terms of x. (c) Give interpretations of the slope and then x and y intercepts of N(x) if possible. (d) Calculate and interpret N 1 (500). (e) An internal audit reveals that the profit made by the company of the sale of 10 units of its product, before advertising costs have been accounted for, is $2,000. What are the implications regarding the company s advertising campaign? Explain. 59. Write a paragraph about each of the following: (a) The end behavior of polynomials and rational functions. (b) Shapes of graphs of power functions (positive, negative integer, fractional power). 60. The formula K(C) = C converts Celsius temperature to Kelvin. The formula C(F ) = 5 9 (F 32) converts Fahrenheit temperature to Celsius. (a) Write a composite function that will convert Fahrenheit temperature to Kelvin. (b) Convert the boiling point of water (212 F) and the freezing point of water (32 F) to Kelvin. 61. Consider the following functions. Which one(s) are not invertible and why? For those that are, find their inverse functions. Then graph one such pair (function and its inverse) on the same screen of your calculator, copy the graphs to the paper, and explain why the graphs show that these functions are inverse to each other. (a) f(x) = 2 ln(1/x) (b) g(x) = x 2 4x + 5 x (c) h(x) = x + 1 (d) u(x) = x Consider the function f(x) = x2 4 x 3 + 4x 2. Find all its intercepts and asymptotes algebraically, then graph the function without using a calculator. Label all significant points. 8

9 63. Sketch the graph of the polynomial y = 2(x + 3) 3 (x 2) 2. Label all significant points. 64. Sketch the graph of a rational function that has the vertical asymptotes x = 3 and x = 2, horizontal asymptote y = 1 and x-intercepts 0 and A population of animals varies sinusoidally between a low of 700 on February 1 and a high of 900 on August 1. (a) Graph the population against time. (b) Find a formula for the population as a function of time, t, measured in months since the start of the year. (c) According to your formula, what is the population on May 1? 66. An economist consulted by your temporary employment agency indicates that the demand for temporary employment (measured in thousands of job applications per week) in your county can be modeled by the function d = 4.3 sin[0.82(t + 0.3) + 7.3], where t is time in years since January, Find the amplitude, the vertical shift, the horizontal shift, the angular frequency, and the period of this function, and interpret the results. 67. Use a graph to decide whether each expression below is true for all x or not. If it is, prove it algebraically, and if not, find a value of x for which it fails. sin x (a) sin x + cos x = tan x 1 + tan x (b) cos(x 2 ) = (cos x) Simplify each expression as much as possible: (a) 1/ sec x (b) 5 cos 2 x + 5 sin 2 x (c) (1 cos 2 t)/ sin t (d) tan x cos x csc x. 69. What do the graphs of f(x) = (x 4) 2 and g(x) = (x + 1) 3 look like? How do multiple zeros manifest themselves on the graph of a function? 70. Find a possible formula for fourth degree polynomial g that has a double zero at x = 3, g(5) = 0, g( 1) = 0 and g(0) = Give a rough sketch (by hand) of the following polynomials: (a) (x 1)(x 2)(x + 3) (b) ( 2x + 1)(x + 1)(x 2)(x + 2) (c) (x 1) 2 (x + 1) 2 (d) (x + 1)(3x 4)(3 x) 72. Find all zeros and asymptotes of the rational functions below and determine the long term behavior of each one. a. x + 2 x 2 9 b. x 2 2x 3 2x 2 7x + 5 c. 2x3 + 5x 2 + x 2 3x d. x x 2. 9

10 73. Find a possible formula for the function given by a table below: x y undefined Sketch a possible graph for a function with the following properties: f(2) = 3, f has a vertical asymptote x = 0, the horizontal asymptote y = 0 when x and the horizontal asymptote y = 2 when x Find the domain of each of the following functions: f(x) = ln x g(x) = ln(x 2 ) h(x) = ln(x 2 + 3) u(x) = e 2 ln x w(x) = e x The period between two high tides at the ocean coast is 14 hours. High tide is 15 m above sea level, and low tide is 7 m above. Write a formula that models the height of the water at time t, if high tide occurs at t = It costs a company $10,000 to begin production of a new environment-friendly toothbrush, plus $1 for each unit produced. Let x be the number of units produced by the company. (a) Find a formula for C(x), the total cost of making x toothbrushes. (b) Find a formula for the company s average cost per unit, a(x). (c) Graph y = a(x) for 0 < x 50, 000, 0 y 10. Label the horizontal asymptote. (d) Explain in economic terms why the graph of a has the long term behavior that it does. (e) Explain in economic terms why the graph of a has the vertical asymptote that it does. (f) Find a formula for a 1 (y). Give an economic interpretation of a 1 (y). 78. Find all intercepts and asymptotes of the function y = x2 + 5x + 4 x 3 4x. Sketch the graph WITHOUT a calculator; label all significant points. 10